Explanation Result used: (1) Ratio test : Let ∑ k = 1 ∞ a k be an infinite series and let r = lim k → ∞ | a k + 1 a k | . If r < 1 , then the series ∑ k = 1 ∞ a k is converges. (2) The p- series converges if p > 1 and diverges if p ≤ 1 . (3) The power series representation of ln ( 1 − x ) = − ∑ k = 1 ∞ x k k Given: The function is f ( x ) = ln 4 − x 2 . Calculation: The power series representation for f ( x ) = ln 4 − x 2 is obtained as follows, ln 4 − x 2 = 1 2 ln ( 4 − x 2 ) = 1 2 ln 4 ( 1 − x 2 4 ) = 1 2 ln 4 + 1 2 ln ( 1 − x 2 4 ) ( ∵ ln a b = ln a + ln b ) = ln 2 + 1 2 ln ( 1 − x 2 4 ) Substitute x 2 4 for x in ln ( 1 − x ) = − ∑ k = 1 ∞ x k k , ln ( 1 − x 2 4 ) = − ∑ k = 1 ∞ ( x 2 4 ) k k = − ∑ k = 1 ∞ x 2 k k 4 k Therefore, f ( x ) = ln 2 − 1 2 ∑ k = 1 ∞ x 2 k k 4 k . Therefore, the power series representation for f ( x ) = ln 2 − 1 2 ∑ k = 1 ∞ x 2 k k 4 k . Let a k = x 2 k k 4 k . Then, a k + 1 = x 2 ( k + 1 ) ( k + 1 ) 4 ( k + 1 ) . Use the Ratio test to compute the value | a k + 1 a k |

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 11.2, Problem 61E

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To find: The power series representations centered at 0 and find the interval of convergence for the resulting series.

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Chapter 11.2, Problem 60E

Chapter 11.2, Problem 62E

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Chapter 11 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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The power series representations centered at 0 and find the interval of convergence for the resulting series.

Solution Summary: The author explains the power series representations centered at 0 and find the interval of convergence for the resulting series.

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To find: The power series representations centered at 0 and find the interval of convergence for the resulting series.

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Chapter 11 Solutions